A Collection of Formulæ for the Area of a Plane Triangle

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134 BAKER. FORMUL/E FOR THE AREA OF A PLANE TRIANGLE

troids and axoids (Liovvillc's Journal, XVI, 9-129; 286-336). The applica-
tion of these method.s to the .study of Mechanisms was first made systematically
by Willis (1841); his processes were improved by Rankine (^Machinery and
Milkvork, 187 1); and the whole subject was reformed and freshly stated by Reu-
leaux {Kinematik, 1874), and has since been still further developed by Grashot
and others. Peculiar interest attaches to the "tram-motion" from its furnishing
to Leonardo da Vinci his famous discovery of the elliptic chuck fi)r turning ovals
on the lathe (Chasles, Apergti 531). — W. M. 7.]

A COLLECTION OF FORMULA FOR THE AREA OF A PLANE TRIANGLE.*
By Mr. Marcus Baker, Washington, D. C.

In April, 1883, Mr. James Main, formerly of the U. S. Coast and Geodetic
Survey, published in the Mathematical Magazine d^ collection of forty-six (46) ex-
pressions for the area of a plane triangle, prefacing it with the remark that this
collection "may be regarded as a matter of curiosity," and that about one-half
of the formulae are well known.

In the following August M. Ed. Lucas reprinted this collection in Mathesis
in a classified form, separating the formulae into five groups and adding one for-
mula not contained in Mr. Main's list. The collection has also been reprinted in
the third number of the Tidsskrift for Mathematik, 1883. Some two or three
additional formulae have since been printed in various mathematical publications.

The terms in which the area is expressed in Mr. Main's collection are angles,
sides, perpendiculars, and radii of inscribed, escribed, and circumscribed circles.
No formulae are given involving medians or bisectors. In numbering Mr. Main
has not counted those formulae as distinct which arise from merely permuting the
letters, nor has he in every case given all the forms possible to be obtained by
permuting the letters, though he has generally done so. As numbered, then, he
counts forty-six formulae, but if every form be counted as a distinct one the total
number is ninety-four.

M. Lucas, by making all possible permutations and adding one new form,
makes the number [39, to which some two or three have been added since.

As the matter has proved of interest, the following collection has been
made, which is a still further extension ; the additional formulae being chiefly due
to introducing the medians and bisectors, not used in the former collections. In
this collection Mr. Main's mode of numbering has been followed and formulae

♦Read before the Matliematical section of the philosophical Society of Washington, January 7, 1885.

BAKER. FORMUL.-?: FOR THE AREA OF A PLANE TRIANGLE. 1 35

derived by permutation are not enumerated as distinct formulae. Moreover, for-
mulae e.Kpressed in the same terms, but in different form, are also considered as
but one. For example, in the former lists we find

J = 2R? sin A sin B sin C,

J = l/e^ (sin 2A + sin 2B + sin 2(f),

and J = |/?- [sin' A cos{B — C) + sin» B cos {C — A) + sin' C cos {A — B)]

given as three distinct formula whereas they are here counted as o»e, and the
principle involved herein is employed throughout.

The total number of formulae for the area of a plane triangle in this collec-
tion is ninety-three, not counting those arising from permutation. If these be
counted as distinct the total number is two hundred and sixty-nine.

Owing to pressure of other duties and consequent lack of time some of the
groups in this collection are not so fully worked up as had been planned.

It may be noted that a number of curious and interesting theorems may be
obtained by equating different expressions for the area and reducing. In this
collection we have classified, for convenience, all expressions for the area into five
groups and each group into two parts.

Group I contains formulae which M. Lucas has called unique, i. e. formulae
which do not admit of other similar formula; by merely permuting the letters.
In such formulae all the sides, all the perpendiculars, all the medians, etc. must
enter if one enters.

Group II contains formulae which admit of tivo similar expressions by per-
mutation giving three of a kind.

Group III contains formulae which admit of three similar expressions by per-
mutation giving four of a kind.

Group IV contains formulae which admit oi five similar expressions by per-
mutation giving six of a kind.

Group V contains formulae which admit o{ eleven similar expressions by per-
mutation giving twelve of a kind.

Each group is divided into tivo parts, the first containing and the second not
containing trigonometrical functions. In addition to the foregoing a group of
miscellaneous expressions, not falling within the classification used, has been
added, and called the Miscellaneous Group.

The notation used is as follows :

J ^ the area of the triangle ;
A, B, C ^ the angles ;
a, b,c = the sides opposite A, B, and C respectively
.f =: semi perimeter ^ \ (a -\- b -\- c)\

136 BAKER. FORMULAE FOR THE AREA OF A FLAXE TRIANGLE.

^, >", >'a, >'h, >'c = the radii of circumscribed, inscribed, escribed circles respectively-
//„, hi, h^ = the perpendiculars from A, B, and C respectively ;
III,,, 111,,, III,. = the medians from A, B, and C respectively ;

,''«, t^in fie = the bisectors (internal) of the angles A, B, and C respectively ; and
a = i(/«„ + Wj + ;«,).

GROUP I. PART I.
J =

I. V s{s — a) [s — b)[s — c) = y 2rtV + 2/)^r + zrd- — a^ — b* — c^

2. f l-' <T ((T — ;«„) {a — ?«j) (<T — ;«,)

^1 zni^-niif + 2ini;in'^ + 2;«/w?/ — 111^ — nib — '«/

I -^

= //:VA/ -

VlhJh-ihJh-VhX)X—hJh^-h^^^

abc

2
2

-^ l/(2;«/ + 2Wj2 — w/) (2;«j2 4- 2;«/ — ;«/) (2w/ + 2;«,.2 — w^^)

/' |(^a + n) (^6 + n)(^c + r„)
2\ ^

10,

II.

\f''abchJhK

abc _ /^(r,. + ;-,,)(; -,, + r,,)(r,. + r,,) _ { a' -\- b^ + c^ 1 *

"1 I , I ■ I I l

! -2 + -2 + — 2 + ,;-2

\J 'a f b > e \

n + I'c — r abc 1 I I . I if

— , -i- L '

yliim,,^ + m,:- + m^) ^ [^^^ + ^^ + ^^j

I

12. 2^:^ '^^'- = 1^ (/a.a„ + hA + /.//„)

BAKER. FORMULA FOR THE AREA OF A PLANE TRIANGLE. 1 37

GROUP L PART I. — Contimud.

;3„M a^ b b^ c c ^ a i,,,fiii i

,+

+

V" 2m^ + 2;«/ — ;«/ l/2;«j^ + 2w/ — m.^ V 2;«/ + 2m^ — m^

l/2w„^4- 2m^ — ni^ ^- V 2nh' + 2;«/ — m,^ + i/2;«/ + 2w„2 — w^,/

i6. i-yjiAAi^c

/'a + /i^;, + /«-■ —

I

I I I

K + /r, + K, J

17-

fylMA j T-^+ -^-^- +

I I ' I I ' I I r

.-+- -+- -+-

'8- ^\r?,M. [(« + ^ + c) [^ + I + -^] - i]

,Q 1 f^:^A (« + ^) (^ + ^) (^ + ^)

hjiji,
2>abc

21- -t:i:.kr.^n+r,-rf.

GROUP L PART H.

22. 2P?sm A sin .5 sin C

= ^i'?' (sin 2A -{■ sin 2.5 + sin 2C")

= 1^2 [sinM cos (.5 — C) +sin«^cos(<r— ^) +sin'C"cos {A — B)']

23. ^./? (a cos A + b cos 5 + <: cos C)

= R[a cos (7 cos ^ + ^ cos A cos .5 + t: cos B cos C]

fl2 4. ^ + ^

24. i(rt^cot^4-^'cot.5 4-<^cotC)=^.

cot yi 4- cot .g 4- cot C

138 HALSTED. DEMONSTRATION OF DESCARTES's THEOREM AND EULER'S THEOREM.

GROUP 1. PART II. — Continued.

J =

2c * 1 ^/ + m,' + in?
^' ~^ cot A + coi B -\- cot C

26. V2R,\fi,;i,[i~c<MA^—'By+'c^sjB ~-:rcY+'cos ( C — "^)]

= 2i/R,3Jjl sin (.4 + ^)sm{B +JC) sin (C + ^A)
27. Rf^/iJi,Jic sin A sin B sin C

28. ^R sin ^ sin i? sin C -.- + 7- +

29.

30.

31-

k, + //, + A,.

fcos^

|-/4 cos^B cos^-C

;4 ;3t

, 9,sin(C + |-^) + ,3^,sin(yJ + |5) J^,5, sm (5 + J- (f )
f cos i-A cos 45 cos 5 C

V pa t% /',

s{a — i>){^ — c) {c — a)

[to be continued].

DEMONSTRATION OF DESCARTES'S THEOREM AND EULER'S THEOREM.
By Prof. G. B. H.\lsted, Austin, Texas.

descartes's theorem.
Cutting by diagonals the faces not triangles into triangles, the whole surface
of any polyhedron contains a number of triangular faces four less than double
the number of summits.

Proof.
For, joining all the summits by a single closed broken line, this cuts the
surface into two skew polygons, each of which contains 5 — 2 triangles, where
5 is the number of summits.

*Tidsskrift for Malhematik, 8° Copenhagen, 1883, fifth series, first year, No. 4, p. 136.